For the 2. problem, construct the inscribed sphere. The number of each edge is the area of the triangle built by the edge and one of the two points where the sphere touches one of the adjacent faces.

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(press several times Z-undo and then X-redo to see the steps)

]]>1. Let a, b, c be the sides, p=a+b+c. Set values to (p-a)/2, (p-b)/2, (p-c)/2. The sum of, say, the first two is exactly c.

2. Same idea. Too lazy for a school grad formula, but can easily imagine this with integrals. For each edge PQ assing \int_{[P,Q]}dm, where m denotes the measure over the manifold consisting of that edge. ]]>